The same applies to chords of 3 semitones. The following table plots sonority size downwards on the left against semitone count to the right plus the quality of ancohemitonia denoted with letter A versus cohemitonia denoted with letter C. In general, ancohemitonic combinations are fewer for a given chord or scale size, but used much more frequently so that their names are well known.
Column "0" represents the most commonly used chords.
Quantification of hemitonia and its relationship to ancohemitonia
Column 1 represents chords that barely use the harmonic degrees that column "0" avoids. Column 2, however, represents sounds far more intractable. Column "2C", row "7", another local minimum, refers to the Neapolitan major scale , which is cohemitonic and somewhat less common but still popular enough to bear a name. As music tends towards increasing dissonance through history, perhaps someday Column 2 will be as acceptable as even Column 1 might be, and Column 3 will finally have a place in the harmony of the world. Note, too, that in the highest cardinality row for each column before the terminal zeros begin, the sonority counts are small, except for row "7" and the "3" columns of all sorts.
This explosion of hemitonic possibility associated with note cardinality 7 and above possibly marks the lower bound for the entity called "scale" in contrast to "chord". As shown in the table, anhemitonia is a property of the domain of note sets cardinality 2 through 6, while ancohemitonia is a property of the domain of note sets cardinality 4 through 8 3 through 8 for improper ancohemitonia including unhemitonia as well. This places anhemitonia generally in the range of "chords" and ancohemitonia generally in the range of "scales". The interrelationship of semitones, tritones, and increasing note count can be demonstrated by taking five consecutive pitches from the circle of fifths ;  starting on C, these are C, G, D, A, and E.
Transposing the pitches to fit into one octave rearranges the pitches into the major pentatonic scale : C, D, E, G, A. This scale is anhemitonic, having no semitones; it is atritonic, having no tritones. In addition, this is the maximal number of notes taken consecutively from the circle of fifths for which is it still possible to avoid a semitone. This scale is hemitonic, having a semitone between B and C; it is atritonic, having no tritones.
In addition, this is the maximal number of notes taken consecutively from the circle of fifths for which is it still possible to avoid a tritone. Adding still another note from the circle of fifths gives the major heptatonic scale: C D E F G A B when the fifth is added from below the tonic. This scale is strictly ancohemitonic, having 2 semitones but not consecutively; it is tritonic, having a tritone between F and B.
Past this point in the projection series, no new intervals are added to the Interval vector analysis of the scale,  but cohemitonia results. Similar behavior is seen across all scales generally, that more notes in a scale tend cumulatively to add dissonant intervals specifically: hemitonia and tritonia in no particular order and cohemitonia not already present.
Book From Polychords To Polya Adventures In Musical Combinatorics
While also true that more notes in a scale tend to allow more and varied intervals in the interval vector , there might be said to be a point of diminishing returns , when qualified against the also increasing dissonance, hemitonia, tritonia and cohemitonia. It is near these points where most popular scales lie. Though less used than ancohemitonic scales, the cohemitonic scales have an interesting property. The sequence of two or more consecutive halfsteps in a scale presents the opportunity to "split" the scale by placing the tonic note of the scale on the middle note of the halfstep span.
This allows a leading tone from below resolving upwards, as well as a descending flat-supertonic upper neighbor , both converging on the tonic. The split turns a weakness - dissonance of cohemitonia - to a strength: contrapuntal convergence on the tonic. It is very common that a cohemitonic or even hemitonic scale e.
Western music's system of key signature is based upon the assumption of a heptatonic scale of 7 notes, such that there are never more than 7 accidentals present in a valid key signature. The global preference for anhemitonic scales combines with this basis to highlight the 6 ancohemitonic heptatonic scales,  most of which are common in romantic music , and of which most Romantic music is composed:. Adhering to the definition of heptatonic scales, these all possess 7 modes each, and are suitable for use in modal mutation.
The following lists the key signatures for all possible untransposed modes of the aforementioned heptatonic scales using the note C as the tonic. Images, videos and audio are available under their respective licenses. Home FAQ Contact.
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